#132080

order	512
nilpotency class	4
isoclinism	isoclinic class of #132079
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8459
commutator	#117	C₂²×C₄
Frattini, cosocle	#405	C₂³
prime radical	#540	C₂×C₄
LCS	G ▷ #117 ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #405 ◁ G
LECS	G ▷ #405 ▷ C₂×C₄ ▷ C₂ ▷ C₁

489 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 489 conjugacy classes of subgroups — too many to draw. Only its 78 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,6,12,5,11,25,42,7,13,28,45,26,9,35,37)(3,14,17,48,16,47,53,64,18,39,54,62,24,46,27,59)(4,19,21,56,22,41,58,44,23,36,29,60,20,10,38,40)(8,30,33,52,31,55,43,49,34,57,63,15,32,50,61,51);(1,3,7,18)(2,8,13,34)(4,15,23,52)(5,24,26,16)(6,27,28,53)(9,31,11,32)(10,39,41,14)(12,43,45,61)(17,25,54,35)(19,47,36,46)(20,51,22,49)(21,57,29,30)(33,37,63,42)(38,50,58,55)(40,59,44,64)(48,56,62,60);(2,9,13,11)(6,28)(8,31,34,32)(10,19,41,36)(12,42,45,37)(14,46,39,47)(17,54)(21,29)(25,35)(27,53)(30,57)(33,61,63,43)(38,58)(40,60,44,56)(48,64,62,59)(50,55)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,d^2=h,e^2=i,f^2=g,g^2=i,b^a=bd,c^a=ce,d^a=dg,h^a=hi,d^b=dh,e^b=ehi,f^b=fghi,g^b=gi,d^c=dh,f^c=fi,e^d=ei,f^d=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 39276);